Introduction: the mathematics of energy systems.

The urgent need to decarbonize energy systems gives rise to many challenging areas of interdisciplinary research, bringing together mathematicians, physicists, engineers and economists. Renewable generation, especially wind and solar, is inherently highly variable and difficult to predict. The need to keep power and energy systems balanced on a second-by-second basis gives rise to problems of control and optimization, together with those of the management of liberalized energy markets. On the longer time scales of planning and investment, there are problems of physical and economic design. The papers in the present issue are written by some of the participants in a programme on the mathematics of energy systems which took place at the Isaac Newton Institute for Mathematical Sciences in Cambridge from January to May 2019-see http://www.newton.ac.uk/event/mes. This article is part of the theme issue 'The mathematics of energy systems'.

Scheduling of energy storage.

The increasing reliance on renewable energy generation means that storage may well play a much greater role in the balancing of future electricity systems. We show how heterogeneous stores, differing in capacity and rate constraints, may be optimally, or nearly optimally, scheduled to assist in such balancing, with the aim of minimizing the total imbalance (unserved energy) over any given period of time. It further turns out that in many cases the optimal policies are such that the optimal decision at each point in time is independent of the future evolution of the supply-demand balance in the system, so that these policies remain optimal in a stochastic environment. This article is part of the theme issue 'The mathematics of energy systems'.

Estimation of under-reporting in epidemics using approximations.

Under-reporting in epidemics, when it is ignored, leads to under-estimation of the infection rate and therefore of the reproduction number. In the case of stochastic models with temporal data, a usual approach for dealing with such issues is to apply data augmentation techniques through Bayesian methodology. Departing from earlier literature approaches implemented using reversible jump Markov chain Monte Carlo (RJMCMC) techniques, we make use of approximations to obtain faster estimation with simple MCMC. Comparisons among the methods developed here, and with the RJMCMC approach, are carried out and highlight that approximation-based methodology offers useful alternative inference tools for large epidemics, with a good trade-off between time cost and accuracy.

Modelling under-reporting in epidemics.

Under-reporting of infected cases is crucial for many diseases because of the bias it can introduce when making inference for the model parameters. The objective of this paper is to study the effect of under-reporting in epidemics by considering the stochastic Markovian SIR epidemic in which various reporting processes are incorporated. In particular, we first investigate the effect on the estimation process of ignoring under-reporting when it is present in an epidemic outbreak. We show that such an approach leads to under-estimation of the infection rate and the reproduction number. Secondly, by allowing for the fact that under-reporting is occurring, we develop suitable models for estimation of the epidemic parameters and explore how well the reporting rate and other model parameters can be estimated. We consider the case of a constant reporting probability and also more realistic assumptions which involve the reporting probability depending on time or the source of infection for each infected individual. Due to the incomplete nature of the data and reporting process, the Bayesian approach provides a natural modelling framework and we perform inference using data augmentation and reversible jump Markov chain Monte Carlo techniques.